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Document an algorithm by Fukushima #2246

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@pleroy pleroy commented Jul 13, 2019

Also clean up some old files that were still around.

pleroy added 4 commits July 13, 2019 18:37

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\subsection*{Definitions}
Jacobi's nome $q\of{m}$ is defined as a function of the elliptic integral $K\of{m}$ as:
\[
q\of{m} = \exp\of{\frac{-\gp K\of{1 - m}}{K\of{m}}}
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\DefineAs, for ≔.

Same in other places below.

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Use \Pi for the constant π (typeset in roman as a standard symbol, in ISO 80000 style). \gp is 𝜋, e.g., "let 𝜋 be a permutation...".

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Using \DefineAs in a bunch of places, please check that it makes sense.

Changed to use \Pi.

\]
We have: $K\of{m} = K_X\of{m_c} X\of{m_c}$.

An expression for $E\of{m}$ can be obtained from Legendre's relation ([Fuku11a] equation 2.11):
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E, F, K, and Π are standard notation, and should be typeset in roman. Add them to semtex.sty in the vicinity of LegendrePolynomial, e.g.,

newcommand{\EllipticE}{\operatorname{\StandardSymbol E}}

Keep E_X for the ad hoc terms, as these are not standard notation.

Similarly B and D are fine, since they are Fukushima’s rather than a widespread notation.

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Nope, both Fukushima and NIST use italics for E, F, etc. The Jacobi functions, on the other hand, are in roman but there aren't any in this paper.

\]
These formulæ make it possible, by computing a Maclaurin approximation of $B_X^*\of{m}$ and $E_X\of{m}$, to evaluate $B\of{m}$
and $D\of{m}$ for $m$ close to $0$.
\printbibliography
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Cite the code as well?

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Done, kind-of, there appears to be an @Software tag.

&= \frac{1}{m}\pa{X\of{m_c}\pa{K_X\of{m_c} - E_X\of{m_c}} - \frac{1}{2 K_X\of{m_c}}}
\end{align*}
These formulæ provide a means to compute $B\of{m}$ and $D\of{m}$ for $m$ close to $1$. First, a polynomial approximation of $q\of{m_c}$
is computed, whose leading term is of order $m_c / 16$. Then the $\log$ of that approximation is evaluated, yielding $X\of{m_c}$ (this
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Does $\frac {m_c} {16}$ look bad?

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Yeah, it's better that way.

B\of{m} = -\frac{\gp}{m} B_X^*\of{m}
\]
These formulæ make it possible, by computing a Maclaurin approximation of $B_X^*\of{m}$ and $E_X\of{m}$, to evaluate $B\of{m}$
and $D\of{m}$ for $m$ close to $0$.
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Consider adding a one-sentence conclusion comparing this method with the one described by Fukushima, so that we understand why this is done that way.

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Added a "Conclusion" section.

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As discussed, clarify the conclusion to actually explain the differences between the two methods.

@eggrobin eggrobin added the LGTM label Jul 26, 2019
@pleroy pleroy merged commit 4151ca3 into mockingbirdnest:master Aug 5, 2019
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